System and method for damping motion of a wind turbine

ABSTRACT

A system ( 40 ) for damping motion of a wind turbine ( 10   a ) is provided. The system ( 40 ) includes a sensor ( 42 ), a movable mass ( 44 ), an actuator ( 58 ), and a controller ( 46 ). The sensor ( 44 ) is operable to provide a signal representative of a motion of the wind turbine ( 10   a ) in one or more degree of freedoms. The movable mass ( 44 ) is associated with the actuator ( 58 ) and is disposed on a blade ( 24   a ) of the wind turbine ( 10   a ) and is configured for movement along a length of the blade. In response to the sensor ( 42 ), the controller ( 46 ) is operable to direct the actuator ( 58 ) to move the movable mass ( 48 ) along a length ( 50 ) of the blade ( 24 ) to a degree effective to dampen motion of the wind turbine ( 10   a ) in one or more degree of freedoms.

FIELD OF THE INVENTION

The present invention relates to wind turbines, and more particularly tosystems and methods for damping motion of a wind turbine.

BACKGROUND OF THE INVENTION

Wind turbines continue to garner significant interest in view of thepush for renewable energy worldwide. Typically, wind turbines include arotor having multiple blades, a drive train and a generator housed in anacelle, and a tower. The nacelle and the rotor are typically mounted ontop of the tower. As the interest in wind turbines has developed, so hasthe interest in moving typical land-based wind turbines offshore. Windturbines adapted for offshore (floating wind turbines) environments aimto make use of improved wind conditions and are particularly of interestwhere land is scarce or where land-based regulations are more stringent.Floating wind turbines typically include the same components asland-based wind turbines, but further include a floating platform uponwhich the rotor, nacelle, and tower are disposed. As is readilyappreciated, a number of forces, including wind energy, wave energy, andforces due to the rotation of the rotor's blades will cause movement ofthe floating wind turbine. This movement of the floating wind turbinewhile in operation significantly reduces the efficiency of the floatingwind turbine. Accordingly, improved systems and methods are needed tominimize movement of the floating wind turbine off-shore to achievegreater efficiency.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is explained in the following description in view of thedrawings that show:

FIG. 1 illustrates a typical prior art floating wind turbine.

FIG. 2 illustrates a schematic of the components of a nacelle in theprior art floating turbine of FIG. 1.

FIG. 3 illustrates a front view of the floating wind turbine and showingan X-axis and a Y-axis relative to the wind turbine in accordance withan aspect of the present invention.

FIG. 4 illustrates a floating wind turbine having a system for dampingmotion in accordance with an aspect of the present invention.

FIG. 5 illustrates a rotor blade having a movable mass in accordancewith an aspect of the present invention.

FIG. 6 illustrates another rotor blade having a movable mass inaccordance with an aspect of the present invention.

FIG. 7 illustrates another rotor blade having two movable masses thereonin accordance with an aspect of the present invention.

FIG. 8 is a schematic of a method for operating a wind turbine inaccordance with the present invention.

FIG. 9 illustrates a motion damping system for a wind turbine withinwhich the turbine's motion is approximated as a mass-spring system inaccordance with an aspect of the present invention.

FIGS. 10A-10I show the results of simulating two simultaneouslyresonantly driven systems damping motion in an X and Y direction at thesame time with one movable mass system.

FIGS. 11A-C show the results of an analytic solution used to directmotion of the movable masses in accordance with an aspect of the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

In accordance with one aspect of the present invention, there aredisclosed systems and methods for operating a wind turbine, whichutilize one or movable masses (herein “movable masses”) disposed on oneor more blades of the wind turbine to dampen motion in at least onedegree of freedom. By “on,” it is meant that the movable masses aredisposed on or within the rotor blade of the wind turbine. The systemsand methods described herein are particularly suitable for floating oroffshore wind turbines to dampen an up-down and/or a side-to-side motionof the floating wind turbine. It is understood, however, that thepresent invention is not so limited and that the systems and methodsdescribed herein may be applied as well to land-based wind turbines orother structures having a need for damping motion and/or mitigatingextreme loading events therein.

In accordance with another aspect of the present invention, the movablemasses on the blades act to create driving forces having a phase and amagnitude sufficient to simultaneously dampen oscillations of the windturbine in a corresponding first direction and a second direction, e.g.,an up-down and a side-to-side direction of the wind turbine. In oneembodiment, a phase of the driving forces is determined by an X-Ylocation of the system's center of mass, while a magnitude of thedriving forces is determined by the mass and inertia of the movablemasses. The center-of-mass position for the associated wind turbinesystem may be actively controlled by moving selected ones (one or more)of the movable masses a particular distance (d) from the rotor centeralong an axis the blades as set forth below. The simulated modeldescribed and set forth herein show that the simultaneous damping of themotion of a wind turbine in two degrees of freedom may be achieved byutilizing aspects of the present invention.

Referring to FIG. 1, FIG. 1 illustrates a floating wind turbine as isknown in the art. As is shown, the floating wind turbine 10 rests in abody of water 11 and comprises a buoyant member 12, a floating platform14, a tower 16 mounted on the floating platform 14, a nacelle 18 mountedon the tower 16, and a rotor 20 having a hub 22 and a plurality of rotorblades 24. As shown in FIG. 2, in one embodiment, the nacelle 18comprises a drive shaft 26, a gear box 28 operably associated with thedrive shaft 26, and a generator 30 operably associated with the gear box28. It is understood, however, that the nacelle 18 is not so limited tocontaining these components. For example, in certain embodiments, thenacelle 18 may not include the gear box 28. In operation, the blades 24of the rotor 20 transform wind energy into a rotational motion of thedrive shaft 26. The drive shaft 26 thereafter rotates a rotor (notshown) of the generator 30. The gear box 28 steps up the relatively lowrotational speed of the generator rotor to a more suitable speed for thegenerator 30 to efficiently convert the rotational motion to electricalenergy. Typically, wind turbines comprise three rotor blades 24,although it is understood the present invention is not so limited.

Referring to FIG. 3, there is shown a floating wind turbine 10 a of thetype described above now having a system 40 for damping oscillationsincorporated therein. The system 40 includes movable masses 44 on eachof the blades 24 a as described below. Each movable mass 44 creates acenter of mass imbalance along a length of its associated blade 24 a. Asshown by an exemplary one of blades 24 a in FIG. 3, a center of massimbalance will exist along a first axis 35 extending through the blade24 a. Further, a center of mass imbalance will exist along a second axis37 that is perpendicular to the first axis 35 and which lies in a planeof the rotor 20. By adjusting one or more of the movable masses 44 to apredetermined degree and controlling the center of mass imbalance alongeach axis 35, 37, the center of mass of the system, e.g., floating windturbine 10 a, may be modified to help create driving forces that willsimultaneously dampen oscillations of the wind turbine in acorresponding first direction and a second direction.

When the floating wind turbine 10 a is disposed within a body of water11, the floating wind turbine 10 a will typically oscillate at aspecific frequency in the first direction, e.g., an up-and-down movementof the floating wind turbine along an X-axis 34 as shown bybi-directional arrow A. In addition, it is expected that the floatingwind turbine 10 a will oscillate at a specific frequency in the seconddirection, e.g. side-to-side movement along a Y-axis 36 as shown bybi-directional arrow B. In one embodiment, the X-axis 34 may be definedas a line or axis extending vertically through or parallel to the tower16 and the nacelle 18 and/or may be defined as an axis that isperpendicular to the Y-axis 36. The oscillations along the X-axis 34would be expected at least as a result of buoyant forces acting upon thefloating wind turbine 10 a. The oscillations along the Y-axis 36 wouldbe expected at least due to forces from wind energy and wave energy.

It is understood that aspects of the present invention are not limitedby these definitions of the X and Y axes, but it is critical rather thatthere exists an axis in a first degree of freedom (e.g., along theX-axis 34), a second degree of freedom (e.g., along the Y-axis 36), orboth. As will be further explained herein, aspects of the presentapplication will servo the floating wind turbine 10 back toward areference point, e.g., a reference point 38, at an intersection of theX-axis 34 and the Y-axis 36 using driving forces created by movablemasses on the blades 24.

Referring now to FIG. 4, there is shown more fully the system 40 fordampening oscillations, which may be incorporated into a wind turbine.In one embodiment, the system 40 may be incorporated into an existingwind turbine, such as that shown in FIG. 1. In another embodiment, thewind turbine may initially be manufactured with the system 40 therein.The system 40 within wind turbine 10 a includes sensors 42, movablemasses 44 disposed on at least one of the blades 24 a of the rotor 20 a,and a controller 46 in communication with the sensors 42 and the movablemasses 44. Collectively, the sensors 42, movable masses 44, and thecontroller 46 may provide the predetermined driving forces necessary toquench motion of the wind turbine 10 in two degrees of freedom, e.g.,along the X-axis 34 and the Y-axis 36 as shown in FIGS. 3 and 9. Thesensors 42 comprise one or more sensors for determining an extent ofmovement of the wind turbine 10 in one or more degrees of freedom, e.g.,along the X-axis 34 and the Y-axis 36. Typically, the sensors 42 areconfigured to sense one or more of a frequency, amplitude, and phase ofone or more oscillations of an associated body, e.g., wind turbine 10 a,in one or more degrees of freedom.

In one embodiment, the sensors 42 comprise one or more accelerometersconfigured to measure oscillations of the wind turbine tower 16 and/ornacelle 18, due to a force of wind striking the tower, wave energy, andthe like along the X-axis 34 and the Y-axis 36. In another embodiment,the sensors 42 include or further include gyroscopic sensors to obtain atilted position of the wind turbine 10 a, e.g., a tilted position of thetower 16. In yet another embodiment, the sensors 42 may comprise aglobal positioning system (GPS), which is particularly suitable toobtain a position of the wind turbine along the X-axis 34. For example,the sensor 42 may be configured to determine a magnitude in which areference point on the wind turbine 10 a, e.g., a reference point on thetower 18, lies above sea level at a particular moment in time.

The sensors 42 may be disposed on the wind turbine 10 a at any suitablelocation for determining the oscillations of the wind turbine 10relative to the X-axis 34 and the Y-axis 36. In one embodiment, one ormore sensors 42 are disposed on the tower 16 and the nacelle 18 as shownso as to sense oscillations of the floating wind turbine 10 along theX-axis 34 and the Y-axis 36. Typically, the sensors 42 will convert thesensed accelerations to an electrical signal, signal 43, which may betransmitted to the controller 46 by any suitable wired or wirelessconnection. The signal may be representative of a magnitude and a phaseof motion of the wind turbine 10 in one or more degrees of freedom. Thecontroller 46 will utilize the received information (from the sensors42) representing the movement of the wind turbine 10 in one or moredegrees of freedom to determine (via a forcing function) the extent towhich one or more movable masses 44 in the blades 24 a will be moved todampen motion of the floating wind turbine 10 along the X-axis 34 or theY-axis 36, or both. Via movement of at least one of the movable masses44 associated with the blades 24 a of the rotor 20 a, the system 40 isable to dampen motion of the floating wind turbine 10 a in one or moredegrees of freedom.

The movable masses 44 may be of any suitable size, shape, and masssuitable for the extent of motion to be dampened. One or more of theblades 24 a of the wind turbine 10 a may include a movable mass 44. Inone embodiment, each of the blades 24 a comprises a movable mass 44 asdescribed herein. The movable masses 44 may be disposed on (on orwithin) the blades 24 a in any suitable configuration. In oneembodiment, for example, the movable masses 44 each comprise a fifty(50) kg mass, each which is configured to move a distance (d) along atrack 48 disposed along a length 50, e.g., a longitudinal axis, of theassociated rotor blade 24. Each movement of a movable mass 44 on acorresponding blade 24 a is effective to change a center of mass of thecorresponding blade 24 a. It is understood that for each blade 24 ahaving a movable mass 44, the movable mass 44 may refer to a single bodyor, in another embodiment, to two or more bodies whose masses arecombined for purposes of reference and/or for determining the extent towhich the movable mass 44 will travel along a length of the blade 24 a.The movable masses 44 may move toward or away from a predetermined pointalong the length 50 of its associated blade 24 as instructed by thecontroller 46. For example, in one embodiment, the movable masses 44move the distance (d) away from the blade root 52 of the rotor 20 a.Typically, the movement of the movable masses 44 is relatively linearalong the length 50 of the blade 24 a, but aspects of the presentinvention are not so limited.

In one embodiment, as shown in FIG. 5, an exemplary blade 24 a from thesystem of FIG. 4 is shown as having a body 54 having a length 50 thatextends along a longitudinal axis 56 of the blade 24 a. In addition, theexemplary blade 24 a includes a movable mass 44, the track 48, and anactuator 58 that interfaces or is associated with the movable mass 44.In one embodiment, the actuator 58 is provided on the track 48 and incommunication with the controller 46 and that is operably associatedwith each of the movable masses 44 to move the movable mass 44 adistance (d) along the length of the associated blade 24. The actuator58 may be any suitable pneumatic actuator, hydraulic actuator, motorizedactuator, or other actuator known in the art.

In a particular embodiment, as shown in FIG. 6, exemplary blade 24 acomprises a spar, e.g., an I-shaped spar 60 having a vertical post 62that extends along the length 50 of the corresponding blade 24. A track,e.g., track 48, is disposed along the longitudinal length of theI-shaped spar 60. An exemplary movable mass 44 is disposed on the track48 and is configured to move along the track 48. The actuator 58 isoperably associated with the moveable mass 44 to move the movable mass44 a predetermined distance (d) along the track 48 in response to acommand from the controller 46. In one embodiment, as shown in FIG. 6, amovable mass 44, a corresponding track 48, and the actuator 58 areprovided on one side of the I-shaped spar 60. In another embodiment, asshown in FIG. 7, a movable mass 44, tracks 48, and one or more actuators58 are provided on opposed sides of the I-shaped spar 60. Providing amovable mass 44 on opposed sides of the I-shaped spar 60 as in FIG. 7allows for a more even mass distribution throughout the blade 24. In oneembodiment, the two opposed movable masses 44 are of substantially thesame mass so as to prevent an asymmetric weight distribution to theblade, as well as allowing for a smaller actuator system. The movablemasses 44 on each side of the I-shaped spar 60 may be recognized as asingle mass for reference and for determining the extent to which themovable masses 44 require movement in order to dampen oscillations ofthe associated structure, e.g., floating wind turbine 10.

In another embodiment, the two movable masses 44 each act as anindependent system on a single blade. In one embodiment, a first movablemass 44 is larger in mass than the second movable mass 44. The firstmovable mass 44 may be used for low-frequency drive motion while thesecond smaller mass 44 may be used for high-frequency drive motion. Inyet another embodiment, the first (larger mass) movable mass 44 may beused for a course correction while the second (smaller) movable mass 44may be used for a fine correction. In still another embodiment, a firstand a second movable mass 44 may be substantially identical or identicalin mass as described above. In such an embodiment, the first movablemass 44 could be used for small wave-wind disturbances and the secondmovable mass 44 could be used for large wave-wind disturbances.

Referring again to FIGS. 3-4, the controller 46 is configured to executecomputer readable instructions for establishing a forcing function toquench motion of the floating wind turbine in one or more degrees offreedom. To accomplish this, the controller 46 comprises one or moreinputs for receiving information from the one or more sensors 42.Utilizing the input information and the forcing function, the controller46 is programmed to instruct the actuator 58 to move one or more of themovable masses 44 on the blades to create driving forces sufficient todampen motion in one or more degrees of freedom, e.g., along the X-axis34 and the Y-axis 36. Thus, the extent of movement (distance (d)) of amovable mass 44 on or within each blade 24 a is automated and governedby the controller 46. In one embodiment, the controller 46 is configuredto move selected ones of the movable masses 44 a desired extent along alength of the blades 24 a from the blade root 52 of the track 48. Inaddition, it is contemplated the controller 46 may receive signalsrepresentative of other data necessary to determine the driving forcesnecessary on two coordinate axes to servo the floating wind turbine 10toward a predetermined reference point, e.g., reference point 38. In oneembodiment, for example, the controller 46 may actively stabilize theX-Y position of the floating wind turbine 10 relative to a position ofthe waves or servo to a position of the sea floor.

The controller 46 may comprise, for example, a special purpose computercomprising a microprocessor, a microcomputer, an industrial controller,a programmable logic controller, a discrete logic circuit or othersuitable controlling device. In one embodiment, the controller 46comprises input channels, a memory, an output channel, and a computer.As used herein, the term computer may include a processor, amicrocontroller, a microcomputer, a programmable logic controller (PLC),an application specific integrated circuit, and other programmablecircuits. The memory may include a computer-readable medium or a storagedevice, e.g., floppy disk, a compact disc read only memory (CD-ROM), orthe like. The controller 46 comprises computer readable instructions fordetermining the extent to which one or more movable masses 44 on theblades 24 must be moved to dampen oscillations of the floating windturbine 10 in one or more degrees of freedom, e.g., along the X-axis 34and the Y-axis 36.

In accordance with another aspect of the present invention, there isprovided a method 100 for operating a wind turbine, e.g., floating windturbine 10 a, having a plurality of blades 24 a utilizing the system 40described herein. As shown in FIG. 8, the method comprises step 102 ofgenerating a signal representative of a magnitude and a phase of motionof the wind turbine 10 a in at least one degree of freedom via at leastone sensor 42. The method 100 then comprises step 104 of executing aforcing function in response to the generated signal effective todetermine driving forces necessary to quench the motion of the windturbine in at least one degree of freedom. In one embodiment, the methodfurther comprises step 106 of generating the driving forces by movingmasses 44 disposed on at least one of the plurality of blades 24 a apredetermined distance as determined by the forcing function to quenchthe motion of the wind turbine in at least one degree of freedom. In aparticular embodiment, the motion of the wind turbine is quenched in afirst degree of freedom and a second degree of freedom simultaneously.

It is understood that aspects of the present invention may activelyservo (stabilize) the X-Y position of floating wind turbines. It isunderstood, however, that the systems and methods described herein maybe applied as well to dampen motion or mitigate extreme loading eventsof land-based wind turbines. In the latter case, it would be expectedthat there may be no oscillations in the up-down direction to bedampened, however extreme loading events could be lessened. It is alsonoted that a mass system in the tower 16 of the floating wind turbine 10a, for example, could dampen the up-down motion, while a mass system ina stationary (horizontal) blade would dampen side-to-side motion.However, in a moving system like a floating wind turbine 10 a describedherein, the movable masses 44 have to move in such a way as to havetheir inertial forces properly decompose to the stationary frame (e.g.,the tower and nacelle 18) of the wind turbine 10 a. Accordingly, the X-Yinertial forces from the movable masses 44 should be mathematicallyidentical or substantially identical to the oscillations on the floatingwind turbine 10, for example. These inertial forces in the moving frameare taken into account in the simulation below. As explained above, thecontroller 46 will determine the extent and amount to move the movablemasses 44 on or within one or more of the blades 24 a to create dampingforces sufficient to quench movement of the floating wind turbine 10 a.The following simulation and non-limiting example illustrates that theabove-described systems and methods may be utilized to stabilize theposition of a floating wind turbine for any waves or excited motion.

Example

Coordinate-System Definitions

The coordinate system and definitions used in the simulation of thissystem are set forth below. In this simulation, the turbine's tower andnacelle are modeled as a single mass M, whose vertical and horizontalposition are defined as X (34) and Y (36) respectively. As shown in FIG.9, the rotor 20 a (of the turbine) rotates with an angular velocity Ω inthe θ direction. The rotor 20 a has a mass m_(R) and a mass-moment ofinertia I_(R). Within each blade of the turbine's rotor is a mass 44(m), which is free to move along the interior of the blade at variabledistance (r) from the rotor's center.

Controlled Damping Mechanism

As explained above and shown in the figures, the masses 44 may beindependently moved along their respective axes in a prescribed fashionin order to accomplish the desired effect of creating a pair of drivingforces (in both the X and Y direction) that are resonant with thevertical and horizontal motion of the associated turbine, respectively.FIG. 9 shows the simplified model of a floating turbine system in whichthe turbine's motion, e.g., motion of the floating wind turbine 10 a, isapproximated as a mass-spring system, whose frequencies (ω_(i)) are setby the spring constants k_(X) and k_(Y), and the total mass m_(T) of thesystem: k_(i)=m_(T)ω_(i) ², where i=X and Y.

The fact that the turbine's rotor rotates at a rate (Ω) that isindependent of the frequencies of the turbine's motion (ω_(i)), meansthat a systematic movement of the three movable masses 44 must be foundthat produces driving forces resonant with the turbine's respective X-Ymotion. In one aspect of the present invention, Fourier analysis showsthat by moving the masses along the blade span at a frequencyω_(DR,i)=Ω−ω_(i), the desired effect of creating a driving forceresonant with the turbine's motion is achieved for i=X and Y. A solutioncan be found for the systematic movement of the masses 44, e.g. by thecontroller 46 as described above, that dampens both the X and Y motionsimultaneously (see FIGS. 10A-I and FIGS. 11A-C) such that the(off-resonant) driving force of one direction has little to no effect onthe other direction. This is verified in the simulation set forth below.

Simulation of Mechanics & Dynamics

FIGS. 10A-10I show the results of simulating a controlled,resonantly-driven damping of an initial 10 cm-amplitude turbineoscillation in the X and Y directions. The vertical motion (X) is shownin FIGS. 10A-10C for the center-of-mass position, velocity, andphase-space over the course of the damping sequence. The horizontalmotion (Y) is shown in FIGS. 10D-10F for the center-of-mass position,velocity, and phase-space over the course of the damping sequence. Theazimuthal angle θ and angular velocity are shown in FIGS. 10G and 10Hwhile the X-Y motion of the wind turbine is shown in FIG. 10I. Thespring constants for the X and Y motion were chosen to give motionalperiods larger than the period of the rotor's rotation. As shown inFIGS. 10A-10I, one can see a very clean and constant damping of theturbine's motion over a 2-minute simulation period, in which the turbineis constantly generating its rated power. One should note that therotation rate of the rotor 20 a (and therefore power generated by theturbine) is nearly unaffected by the damping system 40, despite the factthat the masses 44 are moving rapidly within the rotor's interior (arotating frame).

In order to achieve the desired damping, the prescribed motion of themasses was determined analytically, the results of which are shown inFIGS. 11A-11C. Reduced coordinates were used in order to model themotion of the three masses within the blades. Three movable masses 44were decomposed along a two-axis system to give center-of-mass imbalancepositions (“delta's”) for each axis. Analytic solutions were found thatdescribe the prescribed motion that leads to resonantly-driven behaviorof the turbine's center of mass. An exemplary solution is further setforth below in the following sequence of equations. The lines in FIG.11A show the “delta” motion. FIG. 11B shows the position of the threemasses 44 over time. One of the three masses was not required to move,but may simply be biased to some finite value in order to provide areference position. The resulting forcing functions for the X and Ydirections are shown in FIG. 11C.

The simulation used the following values that one would find reasonablefor a practical system to be employed in future wind turbines. The 10 cmoscillation was fully damped in 2 minutes using three masses m=200 kgeach and a range of motion along r of 1-20 m. It is understood that thevalues used here by no means represent rigid values that are incapableof variation; they simply were reasonable enough to make practicalconclusions.

DEFINITIONS Assuming:

M=mass of tower system (platform 12, buoyant member 14, tower 16,nacelle 18)I_(R)=mass moment of inertia of rotorm_(R)=mass of hub and bladesk_(x), k_(y)=spring constant in x and y direction, respectivelym_(i)=fixed mass on blade i, for i=1, 2, 3r_(i)=variable distance of mass hi; from center of rotation

Definition of Energy: Potential (V) and Kinetic (K)

$\mspace{79mu} {V = {\frac{k_{x}}{2}x^{2 +}\frac{k_{y}}{2}y^{2}}}$     K = K_(T) + K_(R) + K_(i)$\mspace{79mu} {{{{where}\mspace{14mu} T} = {{tower}\mspace{14mu} {system}}};{R = {rotor}};{{{and}\mspace{14mu} i} = {{{mass}\mspace{14mu} {i.\mspace{79mu} K_{T}}} = {\frac{M}{2}\left( {{\overset{.}{x}}^{2} + {\overset{.}{y}}^{2}} \right)}}}}$$\mspace{79mu} {K_{R} = {{\frac{M_{R}}{2}\left( {{\overset{.}{x}}^{2} + {\overset{.}{y}}^{2}} \right)} + {\frac{I_{R}}{2}{\overset{.}{\theta}}^{2}}}}$$\mspace{79mu} {K_{i} = {\sum\limits_{t}{\frac{m_{i}}{2}\left( {{\overset{.}{x}}_{i}^{2} + {\overset{.}{y}}_{i}^{2}} \right)}}}$     where:      x_(i) − x = r_(i)cos  θ_(i)     y_(i) − y = r_(i)sin  θ_(i)$\mspace{79mu} {{\overset{.}{x}}_{i} = {\overset{.}{x} + {{\overset{.}{r}}_{i}\cos \; \theta_{i}} - {{\overset{.}{r}}_{i}{\overset{.}{\theta}}_{i}\sin \; \theta_{i}}}}$$\mspace{79mu} {{\overset{.}{y}}_{i} = {\overset{.}{y} + {{\overset{.}{r}}_{i}\sin \; \theta_{i}} + {{\overset{.}{r}}_{i}{\overset{.}{\theta}}_{i}\cos \; \theta_{i}}}}$$\mspace{79mu} {{\theta_{i} = {{\frac{\left( {i - 1} \right)2\pi}{3} + {\theta \mspace{14mu} {for}\mspace{14mu} i}} = 1}},2,3}$$\mspace{79mu} {{\overset{.}{\theta}}_{i} = \overset{.}{\theta}}$$K_{i} = {{\sum\limits_{i}{\frac{m_{i}}{2}\left( {{{{\overset{.}{x}}^{2} + {\overset{.}{y}}^{2} + {{\overset{.}{r}}_{i}^{2}\cos^{2}\theta_{i}} + {{\overset{.}{r}}_{i}^{2}\sin^{2}\theta_{i}} + {r_{i}^{2}{\overset{.}{\theta}}^{2}\sin^{2}\theta_{i}} + {r_{i}^{2}{\overset{.}{\theta}}^{2}\cos^{2}\theta_{i}} + {2\overset{.}{x}{\overset{.}{r}}_{i}\cos \; \theta_{i}} - {2\overset{.}{x}r_{i}\overset{.}{\theta}\sin \; \theta_{i}} + {2\overset{.}{y}{\overset{.}{r}}_{i}\sin \; \theta_{i}} + {2\overset{.}{y}r_{i}\overset{.}{\theta}\cos \; \theta_{i}}} = {\sum\limits_{i}{\frac{m_{i}}{2}\left( {{\overset{.}{x}}^{2} + {\overset{.}{y}}^{2} + {\overset{.}{r}}_{i}^{2} + {r_{i}^{2}{\overset{.}{\theta}}^{2 +}2{{\overset{.}{r}}_{l}\left( {{\overset{.}{x}\cos \; \theta_{i}} + {\overset{.}{y}\sin \; \theta_{i}}} \right)}} - {2r_{i}{\overset{.}{\theta}\left( {{\overset{.}{x}\sin \; \theta_{i}} - {\overset{.}{y}\cos \; \theta_{i}}} \right)}}} \right)\mspace{79mu} {For}\mspace{14mu} {simplicity}}}},{{{assume}\mspace{14mu} m_{1}} = {m_{2} = {{m\; 3} = {{mK} = {{\frac{1}{2}\left( {{\overset{.}{x}}^{2} + {\overset{.}{y}}^{2}} \right)\left( {M + M_{R} + {\sum\limits_{t}m_{t}}} \right)} + {\frac{1}{2}I_{R}{\overset{.}{\theta}}^{2}} + {\frac{m}{2}{\sum\limits_{i}{\overset{.}{r}\left( {{\overset{.}{r}}_{i}^{2} + {r_{i}^{2}{\overset{.}{\theta}}^{2}}} \right)}}}}}}}}} \right)}} + \left( {m{\sum\limits_{i}{{\overset{.}{r}}_{i}\left( {{\overset{.}{x}\cos \; \theta_{i}} + {\overset{.}{y}\sin \; \theta_{i}}} \right)}}} \right) - \left( {m\; \overset{.}{\theta}{\sum\limits_{i}{r_{i}\left( {{\overset{.}{x}\sin \; \theta_{i}} - {\overset{.}{y}\cos \; \theta \; i}} \right)}}} \right)}$${{{where}\mspace{14mu} m_{T}} = \left( {M + M_{R} + {\sum\limits_{i}{mi}}} \right)};{A = \left( {m{\sum\limits_{i}{{\overset{.}{r}}_{i}\left( {{\overset{.}{x}\cos \; \theta_{i}} + {\overset{.}{y}\sin \; \theta_{i}}} \right)}}} \right)};{B = \left( {{- m}\; \overset{.}{\theta}{\sum\limits_{i}{r_{i}\left( {{\overset{.}{x}\sin \; \theta \; i} - {\overset{.}{y}\cos \; \theta \; i}} \right)}}} \right)}$$\mspace{79mu} {A = {{m\overset{.}{x}{\sum\limits_{i}{{\overset{.}{r}}_{i}\cos \; \theta_{i}}}} + {m\overset{.}{y}{\sum\limits_{i}{{\overset{.}{r}}_{i}\sin \; \theta_{i}}}}}}$$\mspace{79mu} {B = {{{- m}\overset{.}{\theta}\overset{.}{x}{\sum\limits_{i}{r_{i}\sin \; \theta_{i}}}} + {m\overset{.}{\theta}\overset{.}{y}{\sum\limits_{i}{r_{i}\cos \; \theta_{i}}}}}}$Note:  sin (θ + a) = sin  θcos a + cos  θsin a  and  cos (θ + a) = cos  θcos a − sin  θsin a

Common terms arise of the form:

$\begin{matrix}{{\sum\limits_{i}{C_{i}\cos \; \theta_{i}}} = {{C_{1}\cos \; \theta} + {C_{2}{\cos \left( {\theta + {2{\pi/3}}} \right)}} + {C_{3}{\cos \left( {\theta + {4{\pi/3}}} \right)}}}} \\{= {{\cos \; {\theta \left( {C_{1} + {C_{2}{\cos \left( {2{\pi/3}} \right)}} + {C_{3}{\cos \left( {4{\pi/3}} \right)}}} \right)}} -}} \\{{\sin \; {\theta\left( {{C_{2}{\sin \left( {2{\pi/3}} \right)}} + {C_{3}{\sin \left( {4{\pi/3}} \right)}}} \right.}}}\end{matrix}$ $\begin{matrix}{{\sum\limits_{i}{C_{i}\sin \; \theta_{i}}} = {{C_{1}\sin \; \theta} + {C_{2}{\sin \left( {\theta + {2{\pi/3}}} \right)}} + {C_{3}{\sin \left( {\theta + {4{\pi/3}}} \right)}}}} \\{= {{\sin \; {\theta \left( {C_{1} + {C_{2}{\cos \left( {2{\pi/3}} \right)}} + {C_{3}{\cos \left( {4{\pi/3}} \right)}}} \right)}} +}} \\{{\cos \; {\theta \left( {{C_{2}{\sin \left( {2{\pi/3}} \right)}} + {C_{3}{\sin \left( {4{\pi/3}} \right)}}} \right)}}}\end{matrix}$${{{{Note}\text{:}\mspace{14mu} {\sin \left( {2{\pi/3}} \right)}};} = \frac{\sqrt{3}}{2}};$$\mspace{70mu} {{{\sin \left( {4{\pi/3}} \right)} = {- \left( \frac{\sqrt{3}}{2} \right)}};}$$\mspace{70mu} {{{\cos \left( {2{\pi/3}} \right)} = {- \left( \frac{1}{2} \right)}};}$$\mspace{70mu} {{\cos \left( {4{\pi/3}} \right)} = {- \left( \frac{1}{2} \right)}}$

Like quantities can be found:

$\begin{matrix}{{C_{1} + {C_{2}{\cos \left( {2{\pi/3}} \right)}} + {C_{3}{\cos \left( {4{\pi/3}} \right)}}} = {C_{1} - {\frac{1}{2}\left( {C_{2} + C_{3}} \right)C_{2}{\sin \left( {2{\pi/3}} \right)}} +}} \\{{C_{3}{\sin \left( {4{\pi/3}} \right)}}} \\{= {\frac{\sqrt{3}}{2}\left( {C_{2} - C_{3}} \right)}}\end{matrix}$

This results in the terms A & B to be written as:

$A = {{m{\overset{.}{x}\left\lbrack {{\cos \; {\theta \left( {{\overset{.}{r}}_{1} - {\frac{1}{2}\left( {{\overset{.}{r}}_{2} + {\overset{.}{r}}_{3}} \right)}} \right)}} - {\sin \; \theta \frac{\sqrt{3}}{2}\left( {{\overset{.}{r}}_{2} - {\overset{.}{r}}_{3}} \right)}} \right\rbrack}} + {m{\overset{.}{y}\left\lbrack {{\sin \; {\theta \left( {{\overset{.}{r}}_{1} - {\frac{1}{2}\left( {{\overset{.}{r}}_{2} + {\overset{.}{r}}_{3}} \right)}} \right)}} + {\cos \; \theta \frac{\sqrt{3}}{2}\left( {{\overset{.}{r}}_{2} - {\overset{.}{r}}_{3}} \right)}} \right\rbrack}}}$$B = {{{- m}\overset{.}{x}{\overset{.}{\theta}\left\lbrack {{\sin \; {\theta \left( {r_{1} - {\frac{1}{2}\left( {r_{2} + r_{3}} \right)}} \right)}} + {\cos \; \theta \frac{\sqrt{3}}{2}\left( {r_{2} - r_{3}} \right)}} \right\rbrack}} + {m\overset{.}{y}{\overset{.}{\theta}\left\lbrack {{\cos \; {\theta \left( {r_{1} - {\frac{1}{2}\left( {r_{2} + r_{3}} \right)}} \right)}} - {\sin \; \theta \frac{\sqrt{3}}{2}\left( {r_{2} - r_{3}} \right)}} \right\rbrack}}}$

Similar terms can be found and are recognized to be center-of-massimbalances δ_(i) caused by the arrangement r_(i) of the three massesm_(i).

DEFINE:

${\delta_{1} = {r_{1} - {\frac{1}{2}\left( {r_{2} + r_{3}} \right)}}};{\delta_{2} = {\frac{\sqrt{3}}{2}\left( {r_{2} - r_{3}} \right)}}$${{\overset{.}{\delta}}_{1} = {{\overset{.}{r}}_{1} - {\frac{1}{2}\left( {{\overset{.}{r}}_{2} + {\overset{.}{r}}_{3}} \right)}}};{{\overset{.}{\delta}}_{2} = {\frac{\sqrt{3}}{2}\left( {r_{2} - r_{3}} \right)}}$${{\overset{¨}{\delta}}_{1} = {{\overset{¨}{r}}_{1} - {\frac{1}{2}\left( {{\overset{¨}{r}}_{2} + {\overset{¨}{r}}_{3}} \right)}}};{{\overset{¨}{\delta}}_{2} = {\frac{\sqrt{3}}{2}\left( {r_{2} - r_{3}} \right)}}$

δ₁=center of mass imbalance along the ‘1’ axis (axis defined by mass #1)(shown as axis 35 in FIG. 3)δ₂=center of mass imbalance along the ‘2’ axis (perpendicular to the ‘1’axis and lying within the rotor plane) (shown as axis 37 in FIG. 3).From here terms A and B can be reduced to the following using thecenter-of-mass imbalance terms:

A=m{dot over (x)}[{dot over (δ)}₁ cos θ−{dot over (δ)}₂ sin θ]+m{dotover (y)}[{dot over (δ)}₁ sin θ+{dot over (δ)}₂ cos θ]

B=−m{dot over (x)}{dot over (θ)}[δ ₁ sin θ+δ₂ cos θ]+m{dot over (y)}{dotover (θ)}[δ ₁ cos θ−δ₂ sin θ]

Call A+B=K_(XT); where “XT”=cross termsThe kinetic energy in the cross terms can then simply be written:

K _(XT) =m({dot over (x)} cos θ+{dot over (y)} sin θ)[{dot over (δ)}₁−δ₂{dot over (θ)}]−m({dot over (x)} sin θ−{dot over (y)} cos θ)[{dot over(δ)}₂+δ₁{dot over (θ)}]

and the Lagrangian can then be written out as (L=K−V):

$L = {{\frac{m_{T}}{2}\left( {{\overset{.}{x}}^{2} + {\overset{.}{y}}^{2}} \right)} + {\frac{I_{R}}{2}{\overset{.}{\theta}}^{2}} + {\frac{m}{2}\Sigma \; {\overset{.}{r}}_{i}^{2}} + {\frac{m}{2}\Sigma \; {\overset{.}{r}}_{i}^{2}{\overset{.}{\theta}}^{2}} + K_{XT} - {\frac{k_{x}}{2}x^{2}\frac{k_{y}}{2}y^{2}}}$∂_(x) L=−k _(x) x; ∂ _(y) L=−k _(y) y; ∂ _(θ) L=m({dot over (δ)}₁−δ₂{dotover (θ)})(−{dot over (x)} sin θ+{dot over (y)} cos θ)−m({dot over(δ)}₂+δ₁{dot over (θ)})({dot over (x)} cos θ+{dot over (y)} sin θ)

∂_({dot over (x)}) L=m _(T) {dot over (x)}+m[({dot over (δ)}₁−δ₂{dotover (θ)})cos θ−({dot over (δ)}₂+δ₁{dot over (θ)})sin θ]

∂_({dot over (y)}) L=m _(T) {dot over (y)}+m[({dot over (δ)}₁−δ₂{dotover (θ)})sin θ−({dot over (δ)}₂+δ₁{dot over (θ)})cos θ]

∂_({dot over (θ)}) L=(I _(R) +mΣr _(i) ²){dot over (θ)}+m[({dot over(x)} cos θ+{dot over (y)} sin θ)(−δ₂)−({dot over (x)} sin θ−{dot over(y)} cos θ)δ₁]

The equations of motion follow:

{circumflex over (x)} direction)d _(t)∂_({dot over (x)}) L−∂ _(x) L=f_(ext,x)

m _(T) {umlaut over (x)}+k _(x) x=f _(ext,x) −m[({umlaut over (δ)}₁−{dotover (δ)}₂{dot over (θ)}−δ₂{umlaut over (θ)})cos θ−({dot over(δ)}₁−δ₂{dot over (θ)}){dot over (θ)} sin θ−({umlaut over (δ)}₂+{dotover (δ)}₁{dot over (θ)}+δ₁{umlaut over (θ)})sin θ−({dot over(δ)}₂+δ₁{dot over (θ)}){dot over (θ)} cos θ]

{circumflex over (y)} direction)d _(t)∂_({dot over (y)}) L−∂ _(y) L=f_(ext,y)

m _(T) ÿ+k _(y) y=f _(ext,y) −m[({umlaut over (δ)}₁−{dot over (δ)}₂{dotover (θ)}−δ₂{umlaut over (θ)})sin θ−({dot over (δ)}₁−δ₂{dot over(θ)}){dot over (θ)} cos θ−({umlaut over (δ)}₂+{dot over (δ)}₁{dot over(θ)}+δ₁{umlaut over (θ)})cos θ−({dot over (δ)}₂+δ₁{dot over (θ)}){dotover (θ)} sin θ]

{tilde over (θ)} direction)d _(t)∂_({dot over (θ)}) L−∂ _(θ) L=T _(ext)

(I _(R) mΣr _(i) ²){umlaut over (θ)}=T _(ext) +m[({dot over (δ)}₁−δ₂{dotover (θ)})(−{dot over (x)} sin θ+{dot over (y)} cos θ)−({dot over(δ)}₂+δ₁{dot over (θ)})(−{dot over (x)} cos θ+{dot over (y)} sinθ)]+m[{dot over (δ)} ₂({dot over (x)} cos θ+{dot over (y)} sinθ)+δ₂({umlaut over (x)} cos θ−{dot over (x)}{dot over (θ)} sin θ+{umlautover (y)} sin θ+{dot over (y)}{dot over (θ)} cos θ)+{dot over (δ)}₁({dotover (x)} sin θ−{dot over (y)} cos θ)+δ₁({umlaut over (x)} sin θ+{dotover (x)}{dot over (θ)} cos θ−{umlaut over (y)} cos θ+{dot over (y)}{dotover (θ)} sin θ)]

And we define: I_(T)=(I_(R)+mΣr_(i) ²)

${\left. {{{{\left. {\left. {{{\left. \hat{X} \right)\mspace{14mu} m_{T}\overset{¨}{x}} + {k_{x}x}} = {{f_{{ext},x} - {m\left\lbrack {{\cos \; {\theta \left( {{\overset{¨}{\delta}}_{1} - {{\overset{.}{\delta}}_{2}\overset{.}{\theta}} - {\delta_{2}\overset{¨}{\theta}} - {{\overset{.}{\delta}}_{2}\overset{.}{\theta}} - {\delta_{1}{\overset{.}{\theta}}^{2}}} \right)}} - {\sin \; {\theta \left( {{\overset{¨}{\delta}}_{2} + {{\overset{.}{\delta}}_{1}\overset{.}{\theta}} + {\delta_{1}\overset{¨}{\theta}} + {{\overset{.}{\delta}}_{1}\overset{.}{\theta}} - {\delta_{2}{\overset{.}{\theta}}^{2}}} \right)}}} \right\rbrack}} = {f_{{ext},x} + {{m\left\lbrack {{\delta_{\bot}{\overset{.}{\theta}}^{2}} + {2{\overset{.}{\delta}}_{2}\overset{.}{\theta}} + {\delta_{z}\overset{¨}{\theta}} - {\overset{¨}{\delta}}_{1}} \right)}\cos \; \theta} + {\left( {{{- \delta_{2}}{\overset{.}{\theta}}^{2}} + {2{\overset{.}{\delta}}_{1}\overset{.}{\theta}} + {\delta_{1}\overset{¨}{\theta}} + {\overset{¨}{\delta}}_{2}} \right)\sin \; \theta}}}} \right\rbrack \hat{y}} \right)\mspace{14mu} m_{T}\overset{¨}{y}} + {k_{y}y}} = {{f_{{ext},y} - {m\left\lbrack {{\sin \; {\theta \left( {{\overset{¨}{\delta}}_{1} - {{\overset{.}{\delta}}_{2}\overset{.}{\theta}} - {\delta_{2}\overset{¨}{\theta}} - {{\overset{.}{\delta}}_{2}\overset{.}{\theta}} - {\delta_{1}{\overset{.}{\theta}}^{2}}} \right)}} + {\cos \; {\theta \left( {{\overset{¨}{\delta}}_{2} + {{\overset{.}{\delta}}_{2}\overset{.}{\theta}} + {\delta_{1}\overset{¨}{\theta}} + {{\overset{.}{\delta}}_{1}\overset{.}{\theta}} - {\delta_{2}{\overset{.}{\theta}}^{2}}} \right)}}} \right\rbrack}} = {f_{{ext},y} + {m\left\lbrack {{\sin \; {\theta \left( {{\delta_{1}{\overset{.}{\theta}}^{2}} + {2{\overset{.}{\delta}}_{2}\overset{.}{\theta}} + {\delta_{2}\overset{¨}{\theta}} - {\overset{¨}{\delta}}_{1}} \right)}} - {\cos \; {\theta \left( {{{- \delta_{2}}{\overset{.}{\theta}}^{2}} + {2{\overset{.}{\delta}}_{1}\overset{.}{\theta}} + {\delta_{1}\overset{¨}{\theta}} + {\overset{¨}{\delta}}_{2}} \right)}}} \right\rbrack}}}}\hat{\theta}} \right)\mspace{14mu} I_{T}\overset{¨}{\theta}} = {{_{ext} + {m\left\lbrack {{\sin \; {\theta \left( {{- {\overset{.}{x}\left( {{\overset{.}{\delta}}_{1} - {\delta_{2}\overset{.}{\theta}}} \right)}} - {\overset{.}{y}\left( {{\overset{.}{\delta}}_{2} + {\delta_{1}\overset{.}{\theta}}} \right)} + {\overset{.}{y}{\overset{.}{\delta}}_{2}} + {\delta_{2}\left( {\overset{¨}{y} - {\overset{.}{x}\overset{.}{\theta}}} \right)} + {\overset{.}{x}{\overset{.}{\delta}}_{1}} + {\delta_{1}\left( {\overset{¨}{x} + {\overset{.}{y}\overset{.}{\theta}}} \right)}} \right)}} + {\cos \; {\theta \left( {{\overset{.}{y}\left( {{\overset{.}{\delta}}_{1} - {\delta_{2}\overset{.}{\theta}}} \right)} - {\overset{.}{x}\left( {{\overset{.}{\delta}}_{2} + {\delta_{1}\overset{.}{\theta}}} \right)} + {\overset{.}{x}{\overset{.}{\delta}}_{2}} + {\delta_{2}\left( {\overset{¨}{x} + {\overset{.}{y}\overset{.}{\theta}}} \right)} - {\overset{.}{y}{\overset{.}{\delta}}_{1}} + {\delta_{1}\left( {{\overset{.}{x}\overset{.}{\theta}} - \overset{¨}{y}} \right)}} \right)}}} \right\rbrack}} = {_{ext} + {m\left\lbrack {{\left( {{\delta_{2}\overset{¨}{x}} - {\delta_{1}\overset{¨}{y}}} \right)\cos \; \theta} + {\left( {{\delta_{1}\overset{¨}{x}} + {\delta_{2}\overset{¨}{y}}} \right)\sin \; \theta}} \right\rbrack}}}$

We can now recognize that the center-of-mass imbalance terms lead toeffective accelerations in the ‘1’ and ‘2’ directions.

a ₁=(δ₁{dot over (θ)}²+2{dot over (δ)}₂{dot over (θ)}+δ₂{umlaut over(θ)}−{umlaut over (δ)}₁) a ₂=(δ₂{dot over (θ)}²−2{dot over (δ)}₁{dotover (θ)}−δ₁{umlaut over (θ)}−{umlaut over (δ)}₂)

The equations of motion can then simply be written below as:

$\left. {{{{\left. {{{{\left. \hat{x} \right)\mspace{14mu} m_{T}\overset{¨}{x}} + {k_{x}x}} = {f_{{ext},x} + {m\left( {{a_{1}\cos \; \theta} - {a_{2}\sin \; \theta}} \right)}}}\hat{y}} \right)\mspace{14mu} m_{T}\overset{¨}{y}} + {k_{y}y}} = {f_{{ext},y} + {m\left( {{a_{1}\sin \; \theta} + {a_{2}\cos \; \theta}} \right)}}}\hat{\theta}} \right)\mspace{14mu} \begin{matrix}{{I_{T}\overset{¨}{\theta}} = {_{ext} + {m\left( {{\left( {{\delta_{2}\overset{¨}{x}} - {\delta_{1}\overset{¨}{y}}} \right)\cos \; \theta} + {\left( {{\delta_{1}\overset{¨}{x}} + {\delta_{2}\overset{¨}{y}}} \right)\sin \; \theta}} \right)}}} \\{= {_{ext} - {m\left( {{\left( {{\delta_{1}\sin \; \theta} + {\delta_{2}\cos}} \right)\overset{¨}{x}} + {\left( {{{- \delta_{1}}\cos \; \theta} + {\delta_{2}\sin \; \theta}} \right)\overset{¨}{y}}} \right)}}}\end{matrix}$

Damping an Oscillation:

Starting with mass in oscillation with amplitude x_(n)

x(t)=x _(o) cos(ω_(o) t)

To damp (or resonantly damp) the oscillation, one applies a drivingforce:

f(t)=f _(o) cos(ω_(o) t)

the solution to the equation of motion:

m{umlaut over (x)}+kx=f _(o) cos(ω_(o) t)

is x(t)=x_(o) cos ω_(o)t+{dot over (a)}t sin ω_(o)t where á is the timederivative of the oscillation's amplitude.

$\overset{.}{x} = {{{- \omega_{o}}x_{o}\sin \; \omega_{o}t} + {\overset{.}{a}\sin \; \omega_{o}t} + {\overset{\prime}{a}\omega_{o}t\; \cos \; \omega_{o}t}}$$\begin{matrix}{\overset{¨}{x} = {{\left( {\overset{.}{a} - {x_{o}\omega_{o}}} \right)\omega_{o}\cos \; w_{o}t} + {\overset{\prime}{a}\omega_{o}\cos \; \omega_{o}t} - {\overset{.}{a}\omega_{o}^{2}t\; \sin \; \omega_{o}t}}} \\{= {{{- x_{o}}\omega_{o}^{2}\cos \; \omega_{o}t} - {\overset{.}{a}\omega_{o}^{2}t\; \sin \; w_{o}t} + {2\overset{.}{a}\omega_{o}\cos \; \omega_{o}t}}}\end{matrix}$$0 = {{f_{o}\cos \; \omega_{o}t} + {\left( {2\overset{.}{a}\omega_{o}\cos \; \omega_{o}t} \right)m}}$${{{NOTE}\text{:}\mspace{14mu} k} = {m\; \omega_{o}^{2}}},{{{and}\mspace{14mu} f_{o}} = {2\overset{.}{a}m\; \omega_{o}}}$

Therefore: f(t)=2mω_(o) á cos ω_(o)tDefine T to be the time to fully dampen the oscillation. The rate ofchange of the amplitude must then be:

$\overset{.}{a} = \frac{x_{0}}{T}$${f(t)} = {\frac{2m\; \omega_{0}x_{0}}{T}\cos \; \omega_{o}t}$

We can then apply this resonant damping technique to the motion of thefloating turbine: We begin by applying resonant drives in order to killthe oscillation:

$\begin{matrix}{{a_{i}(t)} = {a_{oi}\cos \; \omega_{i}t}} \\{\left. \hat{x} \right)\mspace{14mu} \begin{matrix}{a_{{app},x} = {{a_{ox}\left( {^{{\omega}_{x}t} + ^{{- {\omega}_{x}}t}} \right)}/2}} \\{= {{a_{1}\cos \; \Omega \; t} - {a_{2}\sin \; \Omega \; t}}} \\{= {1/{2\left\lbrack {{a_{1}\left( {{e^{\hat{}}{\Omega}\; t} + e^{\hat{}} - {{\Omega}\; t}} \right)} + {\; {a_{2}\left( {{e^{\hat{}}{\Omega}\; t} - e^{\hat{}} - {{\Omega}\; t}} \right)}}} \right\rbrack}}}\end{matrix}} \\\left. \left. {{a_{ox}\left( {^{{\omega}_{x}t} + ^{{- {\omega}_{x}}t}} \right)} = {{\left( {a_{1} + {\; a_{2}}} \right)e^{\hat{}}{\Omega}\; t} + {\left( {a_{1} - {\; a_{2}}} \right)e^{\hat{}}} - {{\Omega}\; t}}} \right) \right\rbrack \\{\left. \hat{y} \right)\mspace{14mu} \begin{matrix}{a_{{app},y} = {{a_{oy}\left( {^{{\omega}_{y}t} + ^{{- {\omega}_{y}}t}} \right)}/2}} \\{= {{a_{1}\sin \; \Omega \; t} + {a_{2}\cos \; \Omega \; t}}} \\{= {1/{2\left\lbrack {{{- }\; {a_{1}\left( {{e^{\hat{}}{\Omega}\; t} - e^{\hat{}} - {{\Omega}\; t}} \right)}} + {a_{2}\left( {{e^{\hat{}}{\Omega}\; t} + e^{\hat{}} - {{\Omega}\; t}} \right)}} \right.}}}\end{matrix}} \\\left. {{a_{oy}\left( {^{{\omega}_{y}t} + ^{{- {\omega}_{y}}t}} \right)} = {{\left( {{{- }\; a_{1}} + a_{2}} \right)e^{\hat{}}{\Omega}\; t} + {\left( {{\; a_{1}} + a_{2}} \right)e^{\hat{}}} - {{\Omega}\; t}}} \right)\end{matrix}$

Note: Typically, Ω>>ω_(y)>ω_(x); This allows us to treat this as aFourier problem in which we envision the fast oscillation Ω as a carrierfrequency (as the masses are moving within this moving reference frame)and we associate the frequency of the resonant drive as an off-resonantsideband. Because of this definition of Ω, the positive sideband will beoff-resonant and not contribute to the motion of the floating turbinesystem.

We will now find the prescribed analytic solution for simultaneouslydamping out motion in both x and y. The center-of-mass imbalances can bewritten as a linear combination of two frequencies ω_(j)(for j=1,2).c_(kj) and b_(kj) represent complex Fourier amplitudes.

δ_(k)(t)=Σ_(j) C _(kj) e ^(iω) ^(j) ^(t) +b _(kj) e ^(−iω) ^(j) ^(t)k=1,2; j=1,2

{dot over (δ)}_(k)(t)=Σ_(j)(iω _(j))C _(kj) e ^(iω) ^(j) ^(t)−(iω _(j))b_(kj) e ^(−iω) ^(j) ^(t)

{umlaut over (δ)}_(k)(t)=Σ_(j)−ω_(j) ²(C _(kj) e ^(iω) ^(j) ^(t) +b_(kj) e ^(−iω) ^(j) ^(t))

δ₁ =c ₁₁ e ^(iω) ¹ ^(t) +c ₁₂ e ^(iω) ² ^(t) +b ₁₁ e ^(−iω) ¹ ^(t) +b ₁₂e ^(−iω) ² ^(t)

δ₂ =c ₂₁ e ^(iω) ¹ ^(t) +c ₂₂ e ^(iω) ² ^(t) +b ₂₁ e ^(−iω) ¹ ^(t) +b ₂₂e ^(−iω) ² ^(t)

Recall:

{circumflex over (x)})a _(ox)(e ^(iω) ^(x) ^(t) +e ^(−iω) ^(x) ^(t))=[e^(i(omega)t)(a ₁ +ia ₂)+e ^(−i(omega)t)(a ₁ −ia ₂)]

{circumflex over (y)})a _(oy)(e ^(iω) ^(y) ^(t) +e ^(−iω) ^(y) ^(t))=i[e^(i(omega)t)(a ₁ +ia ₂)+e ^(−i(omega)t)(a ₁ −ia ₂)]

a ₁=δ₁Ω²+2{dot over (δ)}₂ Ω−{umlaut over (δ)}. a ₂=δ₂Ω²−2{dot over(Ω)}₁Ω−{umlaut over (δ)}₂

We can then rewrite the terms above in terms:

$\left. {\left. {{{a_{1} + {\; a_{2}}} = {{{\delta_{1}\Omega^{2}} - {2{\overset{.}{\delta}}_{1}\Omega} - {\overset{¨}{\delta}}_{1} + {{\delta}_{2}\Omega^{2}} + {2{\overset{.}{\delta}}_{2}\Omega} - {{\overset{¨}{\delta}}_{2}} - {\sum\limits_{j = 1}^{2}{^{{\omega}_{j}t}\left( {{C_{1j}\Omega^{2 -}2{{\Omega}\left( {\omega}_{j} \right)}C_{1j}} + {\omega_{j}^{2}C_{1j}} + {\; C_{2j}\Omega^{2}} + {2{\Omega \left( {\omega}_{j} \right)}C_{2j}} + {{\omega}_{j}^{2}C_{2j}}} \right)}} + {^{{- {\omega}_{j}}t}\left( {{b_{1j}\Omega^{2}} - {2\; {\Omega \left( {\omega}_{j} \right)}b_{1j}} + {\omega_{j}^{2}b_{1j}} + {\; b_{2j}\Omega^{2}} - {2{\Omega \left( {\omega}_{j} \right)}b_{2j}} + {\omega_{j}^{2}b_{2j}}} \right)}} = {{\sum\limits_{j = 1}^{2}{^{{\omega}_{j}t}\left( {{C_{1j}\left( {\Omega^{2} + {2\omega_{j}\Omega} + \omega_{j}^{2}} \right)} + {\; {C_{2j}\left( {\Omega^{2} + {2\omega_{j}\Omega} + \omega_{j}^{2}} \right)}}} \right)}} + {^{{- {\omega}_{j}}t}\left( {{b_{1j}\left( {{\Omega^{2 -}2\omega_{j}\Omega} + \omega_{j}^{2}} \right)} + {\; {b_{2j}\left( {\Omega^{2} - {2\omega_{j}\Omega} + \omega_{j}^{2}} \right)}}} \right)}}}}{{a_{1} + {\; a_{2}}} = {{\sum\limits_{j = 1}^{2}{{^{{\omega}_{i}t}\left( {\Omega + \omega_{i}} \right)}^{2}\left( {C_{1j} + {\; C_{2j}}} \right)}} + {{^{{- {\omega}_{j}}t}\left( {\Omega - \omega_{j}} \right)}^{2}\left( {b_{1j} + {\; b_{2j}}} \right)}}}{a_{1} = {{\; a_{2}} = {{{\delta_{1}\Omega^{2}} + {2{\overset{.}{\delta}}_{1}\Omega} - {\overset{¨}{\delta}}_{1} - {{\Omega}^{2}\delta_{2}} + {2{\overset{.}{\delta}}_{2}\Omega} - {\overset{¨}{\delta}}_{2}} = {\sum\limits_{j = 1}^{2}{^{{\omega}_{j}t}\left( {{C_{1j}\Omega^{2 +}2{\left( {\omega}_{j} \right)}\Omega \; C_{1j}} + {\omega_{j}^{2}C_{1j}} - {{\Omega}^{2}C_{2j}} + {2{\Omega \left( {\omega}_{j} \right)}C_{2j}} + {\omega_{j}^{2}C_{2j}}} \right)}}}}}} \right){^{{- {\omega}_{j}}t}\left( {{b_{1j}\Omega^{2 +}2{\left( {- {\omega}_{j}} \right)}\Omega \; b_{1j}} + {\omega_{j}^{2}b_{1j}} - {{\Omega}^{2}b_{2j}} + {2{\Omega \left( {- {\omega}_{j}} \right)}b_{2j}} + {\omega_{j}^{2}b_{2j}}} \right)}} \right) = {{\sum\limits_{j = 1}^{2}{^{{\omega}_{j}t}\left( {{C_{1j}\left( {\Omega^{2} - {2\omega_{j}\Omega} + \omega_{j}^{2}} \right)} - {\; {C_{2j}\left( {\Omega^{2} - {2\omega_{j}\Omega} + \omega_{j}^{2}} \right)}}} \right)}} + {^{{- {\omega}_{j}}t}\left( {{b_{1j}\left( {\Omega^{2} + {2\omega_{j}\Omega} + \omega_{j}^{2}} \right)} - {\; {b_{2j}\left( {\Omega^{2} + {2\omega_{j}\Omega} + \omega_{j}^{2}} \right)}}} \right)}}$${a_{1} - {\; a_{2}}} = {{\sum\limits_{j = 1}^{2}{{^{{\omega}_{j}t}\left( {\Omega - \omega_{j}} \right)}^{2}\left( {C_{1j} - {\; C_{2j}}} \right)}} + {{^{{- {\omega}_{j}}t}\left( {\Omega + w_{j}} \right)}^{2}\left( {b_{1j} - {\; b_{2j}}} \right)}}$

We can then rewrite the above equations in this reduced form:

${\left. {{{\left. \hat{x} \right)\mspace{14mu} {a_{ox}\left( {^{{\omega}_{x}t} + ^{{- {\omega}_{x}}t}} \right)}}=={{\sum\limits_{j = 1}^{2}{\left( {\Omega + \omega_{j}} \right)^{2}\left( {C_{1j} + {\; C_{2j}}} \right)^{{{({\Omega + w_{j}})}}t}}} + {\left( {\Omega - \omega_{j}} \right)^{2}\left( {b_{1j} + {\; b_{2j}}} \right)^{{{({\Omega - \omega_{j}})}}t}} + {\left( {\Omega - \omega_{j}} \right)^{2}\left( {C_{1j} - {\; C_{2j}}} \right)^{{- {{({\Omega - \omega_{j}})}}}t}} + {\left( {\Omega + \omega_{j}} \right)^{2}\left( {b_{1j} - {\; b_{2j}}} \right)^{{- {{({\Omega + \omega_{j}})}}}t}}}}\hat{y}} \right)\mspace{14mu} \frac{a_{oy}}{i}\left( {^{{\omega}_{y}t} + ^{{- {\omega}_{y}}t}} \right)} = {{\sum\limits_{j = 1}^{2}{\left( {\Omega + \omega_{j}} \right)^{2}\left( {C_{1j} + {\; C_{2j}}} \right)^{{{({\Omega + w_{j}})}}t}}} + {\left( {\Omega - \omega_{j}} \right)^{2}\left( {b_{1j} + {\; b_{2j}}} \right)^{{{({\Omega - \omega_{j}})}}t}} - {\left( {\Omega - \omega_{j}} \right)^{2}\left( {C_{1j} - {\; C_{2j}}} \right)^{{- {{({\Omega - \omega_{j}})}}}t}} - {\left( {\Omega + \omega_{j}} \right)^{2}\left( {b_{1j} - {\; b_{2j}}} \right)^{{- {{({\Omega + \omega_{j}})}}}t}}}$

Again, the positive sidebands (‘Ω+ω_(j)’ terms) will not contribute toany damping, while the negative sidebands (‘Ω−ω_(j)’ terms) will do allof the damping. Therefore call ω_(x)=Ω−ω₁; ω_(y)=Ω−ω₂.

The above can then be written in matrix form, and solved for the c's andb's:

$\begin{matrix}0 & \; & 1 & i & 0 & 0 & 1 & {- i} & 0 & 0 & \; & c_{11} \\0 & \; & 0 & 0 & 1 & i & 0 & 0 & 1 & {- i} & \; & c_{21} \\{a_{ox}/\left( {\Omega - \omega_{1}} \right)^{2}} & \; & 1 & {- i} & 0 & 0 & 1 & i & 0 & 0 & \; & c_{12} \\0 & = & 0 & 0 & 1 & {- i} & 0 & 0 & 1 & i & \; & c_{22} \\0 & \; & i & {- 1} & 0 & 0 & {- i} & {- 1} & 0 & 0 & \; & b_{11} \\0 & \; & 0 & 0 & l & {- 1} & 0 & 0 & {- i} & {- 1} & \; & b_{21} \\0 & \; & {- i} & {- 1} & 0 & 0 & i & {- 1} & 0 & 0 & \; & b_{12} \\{a_{oy}/\left( {\Omega - \omega_{2}} \right)^{2}} & \; & 0 & 0 & {- i} & {- 1} & 0 & 0 & l & {- 1} & \; & b_{22}\end{matrix}$

The solution above can then be inserted into the definition of thecenter of mass imbalances δ_(k).

and δ_(k)(t)=Σ_(j=1) ² C _(kj) e ^(iwjt) +b _(kj) e ^(−iwjt) k=1,2

These imbalances then dictate the motion of the three masses. It isinteresting to note that to fulfill the requirement for thecenter-of-mass imbalances, only two of the three masses need to be inmotion in certain embodiments. The third mass may simply sit idle at apreset location.

While various embodiments of the present invention have been shown anddescribed herein, it will be obvious that such embodiments are providedby way of example only. Numerous variations, changes and substitutionsmay be made without departing from the invention herein. Accordingly, itis intended that the invention be limited only by the spirit and scopeof the appended claims.

1. A system for damping motion of a wind turbine comprising: a sensoroperable to provide a signal representative of a motion of the windturbine in at least one degree of freedom; a movable mass disposed on ablade of the wind turbine, the movable mass configured for movementalong a length of the blade; and an actuator associated with the movablemass for moving the movable mass along the length of the blade; and acontroller communicably associated with the sensor and the actuator;wherein the controller is operable to receive the signal from the sensorand to responsively direct the actuator to move the movable mass alongthe length of the blade to a degree effective to dampen motion of thewind turbine in the at least one degree of freedom.
 2. The system ofclaim 1, wherein the sensor comprises a first sensor configured to sensemotion of the wind turbine in a first degree of freedom of the windturbine and a second sensor configured to sense motion of the windturbine in a second degree of freedom of the wind turbine.
 3. The systemof claim 2, wherein the wind turbine comprises a plurality of blades,wherein selected ones of the plurality of blades comprise the movablemass and the actuator, and wherein the controller is configured toreceive the signal from the first and second sensors and direct movementof the movable mass via the actuator on at least one of the plurality ofblades to a degree effective to dampen motion of the wind turbine in thefirst degree of freedom and the second degree of freedom.
 4. The systemof claim 1, wherein the blade comprises an I-shaped spar having avertical extent and a track extending longitudinally along the verticalextent, and wherein the actuator is configured to move the movable massa predetermined distance along a length of the track.
 5. The system ofclaim 4, wherein the selected ones of the plurality of blades comprise afirst movable mass and a second movable mass disposed on correspondingtracks on opposed sides of the vertical extent of the I-shaped spar, andwherein the first movable mass and the second movable mass are eachconfigured to move a predetermined distance along a length of the track.6. The system of claim 1, wherein the first degree of freedom isrepresentative of a vertical motion of the wind turbine, wherein thesecond degree of freedom is representative of a horizontal motion of thewind turbine, and wherein the controller is configured to move themovable mass on the selected ones of the plurality of blades to a degreeeffective to provide a pair of driving forces on the wind turbine thatare resonant with the vertical motion and the horizontal motion of thewind turbine.
 7. The system of claim 1, wherein the wind turbine is afloating wind turbine.
 8. A method for operating a wind turbine having aplurality of blades, the method comprising: generating a signalrepresentative of an extent and a phase of motion of the wind turbine inat least one degree of freedom via at least one sensor; and executing aforcing function in response to the generated signal effective todetermine driving forces necessary to quench the motion of the windturbine in the at least one degree of freedom.
 9. The method of claim 8,further comprising generating the one or more forces by moving massesdisposed on at least one of the plurality of blades a predetermineddistance as determined by the forcing function to quench the motion ofthe wind turbine in the at least one degree of freedom.
 10. The methodof claim 9, wherein the at least one degree of freedom comprises a firstdegree of freedom and a second degree of freedom, and wherein the motionof the wind turbine is quenched in the first and second degree offreedom simultaneously.
 11. A wind turbine blade for use with a windturbine comprising: a body having a longitudinal axis; a movable massdisposed on the body effective to change a center of mass of the bladeupon movement of the mass along the longitudinal axis; and an actuatorinterfacing with the movable mass and effective to selectively move themovable mass a predetermined distance along the longitudinal axis. 12.The wind turbine blade of claim 11, wherein the body further comprises:an I-shaped spar having a vertical extent; and a track extendinglongitudinally along the vertical extent; wherein the movable mass isconfigured for movement along the track.